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Firmly nonexpansive mappings in classes of geodesic spaces


Authors: David Ariza-Ruiz, Laurenţiu Leuştean and Genaro López-Acedo
Journal: Trans. Amer. Math. Soc. 366 (2014), 4299-4322
MSC (2010): Primary 47H09, 47H10, 53C22; Secondary 03F10, 47H05, 90C25, 52A41
Published electronically: March 26, 2014
MathSciNet review: 3206460
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Abstract: Firmly nonexpansive mappings play an important role in metric fixed point theory and optimization due to their correspondence with maximal monotone operators. In this paper we do a thorough study of fixed point theory and the asymptotic behaviour of Picard iterates of these mappings in different classes of geodesic spaces, such as (uniformly convex) $ W$-hyperbolic spaces, Busemann spaces and CAT(0) spaces. Furthermore, we apply methods of proof mining to obtain effective rates of asymptotic regularity for the Picard iterations.


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Additional Information

David Ariza-Ruiz
Affiliation: Departamento Análisis Matemático, Fac. Matemáticas, Universidad de Sevilla, Apdo. 1160, 41080-Sevilla, Spain
Email: dariza@us.es

Laurenţiu Leuştean
Affiliation: Simion Stoilow Institute of Mathematics of the Romanian Academy, P. O. Box 1-764, RO-014700 Bucharest, Romania
Email: Laurentiu.Leustean@imar.ro

Genaro López-Acedo
Affiliation: Departamento Análisis Matemático, Fac. Matemáticas, Universidad de Sevilla, Apdo. 1160, 41080-Sevilla, Spain
Email: glopez@us.es

DOI: https://doi.org/10.1090/S0002-9947-2014-05968-0
Keywords: Firmly nonexpansive mappings, geodesic spaces, uniform convexity, Picard iterates, asymptotic regularity, $\Delta$-convergence, proof mining, effective bounds, minimization problems
Received by editor(s): March 10, 2012
Received by editor(s) in revised form: May 29, 2012, and September 22, 2012
Published electronically: March 26, 2014
Article copyright: © Copyright 2014 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.